Mediation Considerations

HEI study objectives

Basic idea for mediation study

To understand the pathways, mechanisms, and intermediates through which a treatment affects an outcome.

First part of mediation: total effect

Second part of mediation: decomposition

Basic idea: understand pathways of effects

Basic DAG for Mediation

Basic DAG for Mediation

Quantities of interest

Total effect: \color{red}{E[Y|T,X]=\beta_{0}+\beta_{1}T +\beta_{2}X}

This equation estimates the total effect of the ban: \color{red}{TE=\beta_{1}(T^{*}-T)} where T^{*} is exposure to ban and T is no exposure.

Mediation model

Estimate two regressions:¹ {E[M|T,X]=\beta_{0}+\beta_{1}T +\beta_{2}X} \color{red}E[Y|T,X,M] = \theta_{0} + \theta_{1}T + \theta_{2}M + \theta_{3}TM + \theta_{4}X + \theta_{5}W

Second equation estimates the “Controlled Direct Effect”: \color{red}{CDE=\theta_{1}+\theta_{3}TM}

Key assumptions

What the hell is the CDE?

Interpretation

This effect is the contrast between the counterfactual outcome if the individual were exposed at T=t and the counterfactual outcome if the same individual were exposed at T=t*, with the mediator set to a fixed level M=m.

English:

“By how much would blood pressure change if the policy were implemented and we held PM_{2.5} fixed at m ?“

Ex: Respiratory symptoms, Sleep

‘Poor respiratory symptoms’ = 1 if frequency of any coughing, wheezing, etc. were “most” or “several” days a week.

Data

• 3 waves, complete data on outcome and mediator

Total Effect

logit(Y_{it}) = \alpha^{village}_{v[i]} + \sum_{r=q}^{T} \beta_{r} d_{r} + \sum_{s=r}^{T} \gamma_{s} fs_{t}+ \sum_{r=q}^{T} \sum_{s=r}^{T} \tau_{rt} (d_{r} \times fs_{t})

• \alpha^{village}_{v[i]} = village-level random intercept
• d_{r} = treatment cohort fixed effects
  
• fs_{t} = time fixed effects
  
• \tau_{rt} = cohort-time ATTs¹

Marginal effects

Cohort-specific ATTs

Mediation model

logit(Y_{it}) = \alpha^{village}_{v[i]} + \sum_{r=q}^{T} \beta_{r} d_{r} + \sum_{s=r}^{T} \gamma_{s} fs_{t}+ \sum_{r=q}^{T} \sum_{s=r}^{T} \tau_{rt} (d_{r} \times fs_{t}) \\ + \delta M_{it} + \sum_{r=q}^{T} \sum_{s=r}^{T} \eta_{rt} (d_{r} \times fs_{t} \times M_{it}) where now we have added:

• \delta = conditional effect of mediator
  
• \eta_{rt} = treatment-mediator product terms

Estimates

• Minimal evidence of mediation.

• Proportion explained: PE = \frac{TE - CDE}{TE} = 0.08

Extensions to multiple mediators

Summary

• Mediation analysis aims are part of HEI project.

• Likely to focus mostly on CDEs.

• Tutorials, packages and macros in R, SAS, Stata available.¹

• Recent R package regmedint from Yoshida and Li (2022)

• Implementation with staggered DiD more likely to require manual implementation.